Mathematics for the interested outsider

Equalizers and coequalizers

Let’s consider another construction from set theory. If we have sets and , and functions and , then we can talk about the subset . This is what we want to generalize.

First off, we know that subsets are subobjects, which are monomorphisms. More to the point, we can look at this subset and take its inclusion function . Then we see that . Furthermore, if any other function satisfies then its image must land in . That is, the function must factor through .

So, given arrows and , each from the object to the object in the category , we construct a new category. The objects are arrows satisfying , and a morphism from to is an arrow so that . Now we define the “equalizer” of and to be a couniversal object in this category, if one exists.

To be a bit more explicit, look at this diagram:
The equalizer is the pair so that for any other there is a unique arrow making the triangle commute. We write it as . Since it’s defined by a universal property, the equalizer is unique up to isomorphism when it exists. If it exists for all pairs of morphisms between the same two objects in the category , we say that “has equalizers”.

Now, as indicated above an equalizer is a monomorphism into . Indeed, let’s say we’ve got two arrows and so that . Then clearly , since is the equalizer of and . So there is a unique morphism so that , and both and must be this unique morphism. Since we can cancel on the left of , is a monomorphism.

The dual notion of an equalizer is a coequalizer. This uses the following diagram:

Go through the above discussion of equalizers and dualize it. Describe a category whose universal object will be the coequalizer. Give an interpretation to this diagram. Prove that the coequalizer of two morphisms is an epimorphism. Try to give a description of coequalizers in , or show that does not have coequalizers.

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